Linear Kelvin Wave Predictions in the $z\to 0$ Limit

Abstract

Linear wave theory captures the essential physics of free-surface flows at a fraction of the computational cost of nonlinear and viscous methods, making it attractive for design, real-time control, and surrogate modeling applications. However, the Kelvin Green's function for a translating point-source generates unbounded wave energy in the $z\to 0$ limit, causing both numerical difficulties and physical inconsistencies. This paper develops a modified kernel for flat-ship theory incorporating an elliptic spanwise line integration that naturally resolves this ill-posedness, yielding finite wave energy over the entire free surface. We then present a fast evaluator for both point and line kernels using contour deformation adapted to the non-analytic Kelvin phase, achieving $10^4$-$10^5$ speedup over direct quadrature while preserving the wake asymptotics. Predictions on the most challenging $z=0$ limit demonstrate physically consistent wave patterns and wave resistance trends. An open-source Julia implementation is provided.

Figures (5)

• Figure 1: Schematic of the free surface prediction problem with submerged point source and flat-ship planform.
• Figure 2: Free surface wave cut and spectra downstream of a unit point-source with $z\to 0$ and line-integrated source on $z=0$. The wave cuts on the left are $\zeta(x\mathbin{=}-8,y,z)$. The spectra on the right are $S^*_\zeta=|\zeta(x\mathbin{=}-40,y(t_+),z)|^2$ evaluated along the divergent wave ridge, \ref{['eq:sp']}, with the carrier demodulated via Hilbert transform. The line-integrated results use half-beam $b\mathbin{=}1$ and $q_0\mathbin{=}\frac{2}{\pi}$ and the dashed line is the predicted $S_\zeta\sim k_y^{-3/2}$ decay.
• Figure 3: Error convergence of the partitioned quadrature method applied to $W(-8,y,z)$ on the same wavecuts in figure \ref{['fig:wavecut']}. Left shows the independence of the error as $z\to 0$ for $N=4$ Gauss-Laguerre points, and right shows the trend with $N$ for $z=-0.01$.
• Figure 4: Flat-ship wave elevation $\zeta$ scaled by the spanwise integrated strength $\tfrac{\pi}{2} q_0b$ for $L=5$. Left: full symmetric field for $b=1$. Right: half-field for $b=1/2$ (top) and $b=2$ (bottom) using the same contour levels.
• Figure 5: Flat ship wave resistance coefficient $C_W$ scaled by $q_0^2$ as a function of the Kelvin-scaled planform length $L$ and width $b$.